1. Fundamentals of statistical testing

• Mean


  Sum of all numbers in a set, divided by the number of numbers
• Standard deviation (SD)

  Spread of the data around the mean. Average difference from the mean

  x = the value from the data

  x w/ the line = the mean

  n = number of values

  E symbol = sum of all (the calculations w each of the values)

  
• Greek, Latin and hat symbols

  Greek = populations

  Latin = samples

  Hat = population estimates

  
• Known distributions

  Some shapes are 'algebraically tractable' (i.e. there is a maths formula to draw the line). Y axis = density (worked out by the pre-set formula)
• Normal distribution


  Continuous, unimodal (only one peak in distribution, one mode), symmetrical and bell-shaped. Normal distribution has fixed proportions and a function of two parameters (mean and SD). Key to be noted that not every bell-curved and symmetrical distribution is a normal distribution

  
• Chi-square distribution

  
• t distribution

  
• Beta distribution

  
• Uniform distribution

  
• Area below the normal curve

  approx. 68% is within ±1sd from the mean

  95% is within ±1.96sd from the mean

  99% is within ±2.58sd from the mean

  
• Proportions to probability


  Proportions are always the same in a normal distribution. If we know smth is normally distributed, we are able to know something about the probability
• Working out proportions - standardisation

  Transforming any distribution to one with mean = 0 and sd = 1, aka transforming variables into z-scores. Do this by subtracting each score from the mean and dividing by the standard deviation
• Working out proportions - probability of z-score

  Use z-table or R to see probability of score e.g. z-score -1.75 has a probability of around 4%. So 4% of variables are below this score, and 96% are above
• using R to work out proportions (Charlie social events e.g.)

  Using original scores and distribution properties;

  charlie_events = 57

  pnorm(charlie_events, mean = 127, sd = 40, lower.tail = FALSE)

  Using z-scores and standard normal distribution;

  charlie_z = -1.75

  pnorm(charlie_z, mean = 0, sd = 1, lower.tail = FALSE)
• Critical value


  A value that cuts off a specific proportion of a distribution, e.g. the top 5%
• Working out critical values - Charlie social events e.g.


  Work backwards; find z-score corresponding to probability of 0.95, then transform z-score to og score

  Using r;

  qnorm(p = 0.95, mean = 127, sd = 40)
• Sampling from distributions


  Collect data on variable = randomly sampling from distribution. Many variables come from normal distribution, some may come from other types... e.g.

  Reaction times - log-normal distribution

  Annual casualties due to horse kicks - Poisson distribution (only deals w/ integers, i.e. can't have .4 of a death)

  Passes/fails on exam - binomial distribution
• Sampling more people


  Samples from the same population will be different from each other. If took many samples and each time calculated the mean, they would have their own distribution.
• Sampling distribution (of the mean)


  Distribution of the means of many samples of a particular size. Distribution is normal and centred around the true population mean.
• Central Limit Theorem

  As n gets larger, the sampling distribution of the mean tends towards a normal distribution with population mean